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A force field is given by $F = 3z+5$. Find the work done in moving an object in this force field along the parabola $z = t^2 + it$ from $z = 0$ to $z = 4+2i$.

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I don't understand why conjugate is taken. How do I interpret it?

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In this problem complex numbers are being used as a notation for 2-dimensional vectors.

The dot product of two vectors is defined by $(a,b)\cdot(c,d)=ac+bd$ but it can also be written as $\Re((\overline{a+ib})(c+id)) = \Re((a-ib)(c+id)) = ac+bd$.

The solution is using this trick to write $F\cdot z$ as $\Re(\bar{F}dz)$.

(The notation in the solution seems a little confused to me. It probably should say $\Re\int_C\bar{F}dz$ instead of $\Re\int_C\bar{F}\cdot dz$. Because either you're using dot products, or you're using complex numbers, but not (in this case) both. But that's a minor quibble.)

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